Is there any side effect to mix dexamethasone with diclofenac in same syringe?

I wouldn't say mixing dexamethasone and diclofenac in one syringe will cause side effects ... however, it could definitely have compatibility issues. Use different syringes for each one.

Has your doctor prescribed these to be taken together?

Just be aware that there is potential for interaction between these medicines. Please see link below.

https://www.drugs.com/interactions-check.php?drug_list=810-0,869-0